Topological properties of sets definable in weakly o-minimal structures

Abstract

The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X,Y and S are non-empty definable sets and S is a large subset of X× Y, then for a large set of tuples 〈\overline{a}₁,…,\overline{a}_2^k〉 ∈ X^{2k}, where k=dim(Y), the union of fibers S_{\overline{a}₁}∪…∪ S_{\overline{a}2^k} is large in Y. Finally, given a weakly o-minimal structure ℳ, we find various conditions equivalent to the fact that the topological dimension in ℳ enjoys the addition property.